18-1 Chapter 18 Data Analysis: Correlation and Regression.

18-1

Chapter 18

Data Analysis: Correlation and

Regression

Figure 18.1 Relationship of Correlation and Regression to the Previous Chapters and the Marketing Research Process

Copyright © 2011 Pearson Education, Inc.

Focus of this Chapter

Relationship toPrevious Chapters

• Correlation

• Regression

• Analytical Framework and Models (Chapter 2)

• Data Analysis Strategy (Chapter 15)

• General Procedure of Hypothesis Testing (Chapter 16)

• Hypothesis Testing Related to Differences (Chapter 17)

Chapter 18 - 2

Approach to Problem

Field Work

Data Preparation and Analysis

Report Preparationand Presentation

Research Design

Problem Definition

Relationship to MarketingResearch Process


Figure 18.2 Correlation and Regression: An Overview

Product Moment Correlation(Fig 18.3 & 18.4)(Table 18.1 & 18.2)

Regression Analysis(Fig 18.3 & 18.4)(Table 18.1 & 18.2)

Bivariate Regression(Fig 18.5, 18.6 & 18.7)(Table 18.3)

Multiple Regression(Table 18.4)


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Opening Vignette

Application to Contemporary Issues (Figs 18.8 & 18.9)

International Social Media Ethics

Chapter 18 - 4

Product Moment Correlation

The product moment correlation, r, summarizes the strength of association between two metric (interval or ratio scaled) variables, say X and Y.

It is an index used to determine whether a linear or straight-line relationship exists between X and Y.

As it was originally proposed by Karl Pearson, it is also known as the Pearson correlation coefficient. It is also referred to as simple correlation, bivariate correlation, or merely the correlation coefficient.

Copyright © 2011 Pearson Education, Inc. Chapter 18 - 5

Product Moment Correlation (Cont.)


Division of the numerator and denominator by (n-1) gives

r varies between -1.0 and +1.0.

The correlation coefficient between two variables will be the same regardless of their underlying units of measurement.


Product Moment Correlation (Cont.)


Table 18.1 Explaining Attitudes Towards Sports Cars

Respondent No. Attitude Toward Sports Cars

Duration of Sports Car Ownership

Importance Attached to Performance

1 6 10 3

2 9 12 11

3 8 12 4

4 3 4 1

5 10 12 11

6 4 6 1

7 5 8 7

8 2 2 4

9 11 18 8

10 9 9 10

11 10 17 8

12 2 2 5

Figure 18.3Plot of Attitude with Duration


4.52.25 6.75 11.25 9 13.5

9

3

6

15.75 18

Duration of Car Ownership

Att

itu

de

Product Moment Correlation


= (10 + 12 + 12 + 4 + 12 + 6 + 8 + 2 + 18 + 9 + 17 + 2)/12= 9.333

= (6 + 9 + 8 + 3 + 10 + 4 + 5 + 2 + 11 + 9 + 10 + 2)/12= 6.583

= (10 -9.33)(6-6.58) + (12-9.33)(9-6.58) + (12-9.33)(8-6.58) + (4-9.33)(3-6.58) + (12-9.33)(10-6.58) + (6-9.33)(4-6.58) + (8-9.33)(5-6.58) + (2-9.33) (2-6.58) + (18-9.33)(11-6.58) + (9-9.33)(9-6.58) + (17-9.33)(10-6.58) + (2-9.33)(2-6.58)= -0.3886 + 6.4614 + 3.7914 + 19.0814 + 9.1314 + 8.5914 + 2.1014 + 33.5714 + 38.3214 - 0.7986 + 26.2314 +

33.5714 = 179.6668


= (10-9.33)2 + (12-9.33)2 + (12-9.33)2 + (4-9.33)2

+ (12-9.33)2 + (6-9.33)2 + (8-9.33)2 + (2-9.33)2 + (18-9.33)2 + (9-9.33)2 + (17-9.33)2 + (2-9.33)2

= 0.4489 + 7.1289 + 7.1289 + 28.4089 + 7.1289+ 11.0889 + 1.7689 + 53.7289 + 75.1689 + 0.1089 + 58.8289 + 53.7289= 304.6668

= (6-6.58)2 + (9-6.58)2 + (8-6.58)2 + (3-6.58)2 + (10-6.58)2+ (4-6.58)2 + (5-6.58)2 + (2-6.58)2

+ (11-6.58)2 + (9-6.58)2 + (10-6.58)2 + (2-6.58)2

= 0.3364 + 5.8564 + 2.0164 + 12.8164 + 11.6964 + 6.6564 + 2.4964 + 20.9764 + 19.5364 + 5.8564 + 11.6964 + 20.9764= 120.9168

Thus, r = 179.6668(304.6668) (120.9168)

= 0.9361

Decomposition of Total Variation


r2 = Explained variation

Total variation

= SSxSSy

= Total variation - Error variationTotal variation

= SSy - SSerror

SSy

Table 18.2Calculation of the Product Moment Correlation


Number

Attitude

(Y)

Duration (X)

XX i

2XX i

YY i

2YY i

YYXX ii

1. 6 10 0.667 0.4489 -0.583 0.3364 -0.3886 2. 9 12 2.667 7.1289 2.417 5.8564 6.4614 3. 8 12 2.667 7.1289 1.417 2.0164 3.7914 4. 3 4 -5.333 28.4089 -3.583 12.8164 19.0814 5. 10 12 2.667 7.1289 3.417 11.6964 9.1314 6. 4 6 -3.333 11.0889 -2.583 6.5664 8.5914 7. 5 8 -1.333 1.7689 -1.583 2.4964 2.1014 8. 2 2 -7.333 53.7289 -4.583 20.9764 3.5714 9. 11 18 8.667 75.1689 7.417 19.5364 38.3214 10. 9 9 -10.333 0.1089 2.417 5.8564 -0.7986 11. 10 17 7.667 58.8289 3.417 11.6964 26.2314 12. 2 2 -7.333 53.7289 -4.583 20.9764 33.5714

Mean 6.583 9.333

Sum 304.668 120.9168 179.6668

Calculation of r


)9168.120)(6668.304(

6668.179r

)9962.10()4547.17(

6668.179

9361.0

Decomposition of the Total Variation (Cont.)

When it is computed for a population rather than a sample, the product moment correlation is denoted by , the Greek letter rho. The coefficient r is an estimator of .

The statistical significance of the relationship between two variables measured by using r can be conveniently tested. The hypotheses are:

H0: = 0H1: 0



Decomposition of the Total Variation (Cont.)

The test statistic is:

which has a t distribution with n - 2 degrees of freedom. For the correlation coefficient calculated based on the data given in Table 13.1,

= 8.414and the degrees of freedom = 12-2 = 10. From the t distribution table (Table 4 in the Statistical Appendix), the critical value of t for a two-tailed test and = 0.05 is 2.228. Hence, the null hypothesis of no relationship between X and Y is rejected.

t = r n-21 - r2

1/2

t = 0.9361 12-21 - (0.9361)2

1/2

Figure 18.4 A Nonlinear Relationship for which r = 0

-3 -2 -1 0 1 2 3

..

..

...

0

1

2

3

4

5

6



Regression Analysis

Regression analysis is used in the following ways: Determine whether the independent variables explain a

significant variation in the dependent variable: whether a relationship exists.

Determine how much of the variation in the dependent variable can be explained by the independent variables: strength of the relationship.

Determine the structure or form of the relationship: the mathematical equation relating the independent and dependent variables.

Predict the values of the dependent variable.


Regression Analysis (Cont.)

Regression analysis is used in the following ways: (Cont.) Control for other independent variables when evaluating

the contributions of a specific variable or set of variables.

Regression analysis is concerned with the nature and degree of association between variables and does not imply or assume any causality.

Statistics: Bivariate Regression Analysis Bivariate regression model. The basic regression

equation is Yi = 0 + 1Xi + ei, where Y = dependent or criterion variable, X = independent or predictor variable, 0 = intercept of the line, 1 = slope of the line, and ei is the error term associated with the i th observation.

Coefficient of determination. The strength of association is measured by the coefficient of determination, r 2. It varies between 0 and 1 and signifies the proportion of the total variation in Y that is accounted for by the variation in X.

Estimated or predicted value. The estimated or predicted value of Yi is = a + b x, where is the predicted value of Yi, and a and b are estimators of

0 and 1, respectively. Copyright © 2011 Pearson Education, Inc. Chapter 18 - 20

Statistics: Bivariate Regression Analysis (Cont.)

Regression coefficient. The estimated parameter b is usually referred to as the non-standardized regression coefficient.

Scattergram. A scatter diagram, or scattergram, is a plot of the values of two variables for all the cases or observations.

Standard error of estimate. This statistic, SEE, is the standard deviation of the actual Y values from the predicted values.

Standard error. The standard deviation of b, SEb, is called the standard error.

Y


Statistics: Bivariate Regression Analysis (Cont.)

Standardized regression coefficient. Also termed the beta coefficient or beta weight, this is the slope obtained by the regression of Y on X when the data are standardized.

Sum of squared errors. The distances of all the points from the regression line are squared and added together to arrive at the sum of squared errors, which is a measure of total error, ∑ej2.

t statistic. A t statistic with n - 2 degrees of freedom can be used to test the null hypothesis that no linear relationship exists between X and Y, or H0: β1 = 0, where

Chapter 18 - 22Copyright © 2011 Pearson Education, Inc.

t = bSEb

Conducting Bivariate Regression Analysis:Plot the Scatter Diagram

A scatter diagram, or scattergram, is a plot of the values of two variables for all the cases or observations.

The most commonly used technique for fitting a straight line to a scattergram is the least-squares procedure.

In fitting the line, the least-squares procedure minimizes the sum of squared errors, ∑ej2 .


Refine the Model

Examine the Residuals

Check Prediction Accuracy

Determine the Strength and Significance of Association

Test for Significance

Estimate Standardized Regression Coefficients

Estimate the Parameters

Formulate the General Model

Plot the Scatter Diagram

Figure 18.5Conducting Bivariate Regression Analysis

Chapter 18 - 24Copyright © 2011 Pearson Education, Inc.


Formulate the Bivariate Regression Model

In the bivariate regression model, the general form of a straight line is: Y = 0 + 1X

where Y = dependent or criterion variable

X = independent or predictor variable

0 = intercept of the line

1 = slope of the line

The regression procedure adds an error term to account for the probabilistic or stochastic nature of the relationship:Yi = 0 + 1X + ei

Figure 18.6 Bivariate Regression

Y


β0 + β1X

Estimate the Parameters


Estimate the Parameters (Cont.)


= (10) (6) + (12) (9) + (12) (8) + (4) (3) + (12) (10) + (6) (4)

+ (8) (5) + (2) (2) + (18) (11) + (9) (9) + (17) (10) + (2) (2)

= 917

= 102 + 122 + 122 + 42 + 122 + 62 + 82 + 22 + 182 + 92 + 172 + 22= 1350

Estimate the Parameters (Cont.)


Standardization is the process by which the raw data are transformed into new variables that have a mean of 0 and a variance of 1 (Chapter 13).

When the data are standardized, the intercept assumes a value of 0.

The term beta coefficient or beta weight is used to denote the standardized regression coefficient.

Byx = Bxy = rxy

There is a simple relationship between the standardized and non-standardized regression coefficients:

Byx = byx (Sx /Sy)

Estimate the Standardized Regression Coefficient



t = bSEb

Chapter 18 - 31

01β:1Η

01β:οΗ


18-32


Using a computer program, the regression of attitude on duration of residence, using the data shown in Table 18.1, yielded the results shown in Table 18.2. The intercept, a, equals 1.0793, and the slope, b, equals 0.5897. Therefore, the estimated equation is:Attitude( ) = 1.0793 + 0.5897 (Duration of Car Ownership)

The standard error, or standard deviation of b is estimated as0.07008, and the value of the t statistic as t = 0.5897/0.0700 =8.414, with n - 2 = 10 degrees of freedom.

From Table 4 in the Statistical Appendix, we see that the criticalvalue of t with 10 degrees of freedom and = 0.05 is 2.228 fora two-tailed test. Since the calculated value of t is larger thanthe critical value, the null hypothesis is rejected.

Y



Attitude( ) = 1.0793 + 0.5897 (Duration of Car Ownership)

The standard error, or standard deviation of b is estimated as0.07008, and the value of the t statistic as t = 0.5897/0.0700 =8.414, with n - 2 = 10 degrees of freedom.

From Table 4 in the Statistical Appendix, we see that the critical

value of t with 10 degrees of freedom and = 0.05 is 2.228 fora two-tailed test. Since the calculated value of t is larger thanthe critical value, the null hypothesis is rejected.

Chapter 18 - 33

Using a computer program, the regression of attitude on duration of residence, using the data shown in Table 18.1, yielded the results shown in Table 18.2. The intercept, a, equals 1.0793, and the slope, b, equals 0.5897. Therefore, the estimated equation is:


Determine Strength and Significance of Association

The total variation, SSy, may be decomposed into the variation accounted for by the regression line, SSreg, and the error or residual variation, SSerror or SSres, as follows:

SSy = SSreg + SSres

where


Figure 18.7 Decomposition of the Total Variation In Bivariate Regression

X

Y

Total variation,SSY

}Residual variation,SS RES

} Explained variation,SS REG

Y


To illustrate the calculations of r2, let us consider again the effect of attitude toward the city on the duration of residence. It may be recalled from earlier calculations of the simple correlation coefficient that:


Determine Strength and Significance of Association (Cont.)

The predicted values ( ) can be calculated using the regression equation:

Attitude ( ) = 1.0793 + 0.5897 (Duration of Car Ownership)

For the first observation in Table 18.1, this value is:

= 1.0793 + 0.5897 x 10 = 6.9763.

For each successive observation, the predicted values are, in order, 8.1557, 8.1557, 3.4381, 8.1557, 4.6175, 5.7969, 2.2587, 11.6939, 6.3866, 11.1042, and 2.2587.



Therefore,

= (6.9763-6.5833)2 + (8.1557-6.5833)2

+ (8.1557-6.5833)2 + (3.4381-6.5833)2

+ (8.1557-6.5833)2 + (4.6175-6.5833)2

+ (5.7969-6.5833)2 + (2.2587-6.5833)2

+ (11.6939 -6.5833)2 + (6.3866-6.5833)2

+ (11.1042 -6.5833)2 + (2.2587-6.5833)2

=0.1544 + 2.4724 + 2.4724 + 9.8922 + 2.4724+ 3.8643 + 0.6184 + 18.7021 + 26.1182+ 0.0387 + 20.4385 + 18.7021= 105.9524



= (6-6.9763)2 + (9-8.1557)2 + (8-8.1557)2

+ (3-3.4381)2 + (10-8.1557)2 + (4-4.6175)2

+ (5-5.7969)2 + (2-2.2587)2 + (11-11.6939)2 + (9-6.3866)2 + (10-11.1042)2 + (2-2.2587)2

= 14.9644

It can be seen that SSy = SSreg + SSres . Furthermore,

r 2 = SSreg /SSy

= 105.9524/120.9168= 0.8762



Another equivalent test for examining the significance of the linear relationship between X and Y (significance of b) is the test for the significance of the coefficient of determination. The hypotheses in this case are:

H0: R2

pop = 0

H1: R2

pop > 0



The appropriate test statistic is the F statistic,

which has an F distribution with 1 and n - 2 degrees of freedom. The F

test is a generalized form of the t test (see Chapter 17). If a random

variable is t distributed with n degrees of freedom, then t2 is F distributed

with 1 and n degrees of freedom. Hence, the F test for testing the

significance of the coefficient of determination is equivalent to testing the

following hypotheses:

or

F = SSreg

SSres/(n-2)



From Table 18.2, it can be seen that:

r2 = 105.9522/(105.9522 + 14.9644) = 0.8762

which is the same as the value calculated earlier.

The value of the F statistic is:

F = 105.9522/(14.9644/10) = 70.8027 with 1 and 10 degrees of freedom.

The calculated F statistic exceeds the critical value of 4.96 determined

from Table 5 in the Statistical Appendix. Therefore, the relationship is

significant, corroborating the results of the t test.




Table 18.3 Bivariate RegressionMultiple R .9361

R2 .8762

Adjusted R2 .8639

Standard Error 1.2233

Analysis of Variance

df Sum of Squares Mean Square

Regression 1 105.9522 105.9522

Residual 10 14.9644 1.4964

F = 70.8027 Significance of F = .0000

VARIABLES IN THE EQUATION

Variable b SE b Beta (B) T Sig. T

Duration .5897 .0700 .9361 8.414 .0000

(Constant) 1.0793 .7434 1.452 .1772

Check Prediction Accuracy



Assumptions

The error term is normally distributed. For each fixed value of X, the distribution of Y is normal.

The means of all these normal distributions of Y, given X, lie on a straight line with slope b.

The mean of the error term is 0. The variance of the error term is constant. This

variance does not depend on the values assumed by X. The error terms are uncorrelated. In other words, the

observations have been drawn independently.

Multiple Regression


Statistics Associated with Multiple Regression (Cont.)

Adjusted R2. R2, coefficient of multiple determination, is adjusted for the number of independent variables and the sample size to account for the diminishing returns. After the first few variables, the additional independent variables do not make much contribution.

Coefficient of multiple determination. The strength of association in multiple regression is measured by the square of the multiple correlation coefficient, R2, which is also called the coefficient of multiple determination.


F test. The F test is used to test the null hypothesis that the coefficient of multiple determination in the population, R2

pop, is zero.

The test statistic has an F distribution with k and

(n - k - 1) degrees of freedom. Partial F test. The significance of a partial regression

coefficient , i , of Xi may be tested using an incremental F statistic. The incremental F statistic is based on the increment in the explained sum of squares resulting from the addition of the independent variable Xi to the regression equation after all the other independent variables have been included.



Partial regression coefficient. The partial regression coefficient, b1, denotes the change in the predicted value, Y , per unit change in X1 when the other independent variables, X2 to Xk, are held constant.



Partial Regression Coefficients

First, note that the relative magnitude of the partial regression coefficient of an independent variable is, in general, different from that of its bivariate regression coefficient.

The interpretation of the partial regression coefficient, b1, is that it represents the expected change in Y when X1 is changed by one unit but X2 is held constant or otherwise controlled. Likewise, b2 represents the expected change in Y for a unit change in X2, when X1 is held constant. Thus, calling b1 and b2 partial regression coefficients is appropriate.


Partial Regression Coefficients (Cont.)

It can also be seen that the combined effects of X1 and X2 on Y are additive. In other words, if X1 and X2 are each changed by one unit, the expected change in Y would be (b1+b2).

Extension to the case of k variables is straightforward. The partial regression coefficient, b1, represents the expected change in Y when X1 is changed by one unit and X2 through Xk are held constant. It can also be interpreted as the bivariate regression coefficient, b, for the regression of Y on the residuals of X1, when the effect of X2 through Xk has been removed from X1.


The relationship of the standardized to the non-standardized coefficients remains the same as before:

B1 = b1 (Sx1/Sy)

Bk = bk (Sxk /Sy)

The estimated regression equation is:

= 0.33732 + 0.48108 X1 + 0.28865 X2

orAttitude = 0.33732 + 0.48108 (Duration) + 0.28865 (Importance)


Partial Regression Coefficients (Cont.)

Multiple R .9721

R2 .9450

Adjusted R2 .9330

Standard Error .8597

Analysis of Variance

df Sum of Squares Mean Square

Regression 2 114.2643 57.1321

Residual 9 6.6524 .7392

F = 77.2936 Significance of F = .0000

VARIABLES IN THE EQUATION

Variable b SE b Beta (B) T Sig. T

Importance .2887 .08608 .3138 3.353 .0085

Duration .4811 .05895 .7636 8.160 .0000

(Constant) .3373 .56736 .595 .5668


Table 18.4 Multivariate Regression

A residual is the difference between the observed value of Yi and the value predicted by the estimated regression equation, .

A plot of residuals against time, or the sequence of observations, will throw some light on the assumption that the error terms are uncorrelated.

Examination of Residuals


Examine the histogram of standardized residuals. Compare the frequency of residuals to the normal distribution and if the difference is small then the normality assumption may be reasonably met.

Examine the normal probability plot of standardized residuals. The normal probability plot shows the observed standardized residuals compared to expected standardized residuals from a normal distribution. If the observed residuals are normally distributed, they will fall on the 45 degree line.

Examination of Residuals (Cont.)


Examine the plot of standardized residuals versus standardized predicted values. This plot should be random with no discernible pattern. This will provide an indication on the assumptions of linearity and constant variance for the error term.

Look at the table of residual statistics and identify any standardized predicted values or standardized residuals that are more than plus or minus 3 standard deviations. Values larger than this may indicate the presence of outliers in the data.



Plotting the residuals against the independent variables provides evidence of the appropriateness. Again, the plot should result in a random pattern.

To examine whether any additional variables should be

included in the regression equation, one could run a regression of the residuals on the proposed variables.





SPSS Windows


The CORRELATE program computes Pearson product moment

correlations and partial correlations with significance levels.

Univariate statistics, covariance, and cross-product deviations

may also be requested.

To select these procedures using SPSS for Windows, click:

Analyze > Correlate > Bivariate …

Scatterplots can be obtained by clicking:

Graphs > Scatter … > Simple Scatter > Define

SPSS Windows (Cont.)

REGRESSION calculates bivariate and multiple regression

equations, associated statistics, and plots. It allows for an

easy examination of residuals. This procedure can be run by

clicking:

Analyze > Regression > Linear …

The detailed steps are illustrated using the data of Table 18.1.


1. Select ANALYZE from the SPSS menu bar.

2. Click CORRELATE and then BIVARIATE.

3. Move "Attitude Towards Sports Cars” and “Duration of Car Ownership"

into the VARIABLES box.

4. Check PEARSON under CORRELATION COEFFICIENTS.

5. Check ONE-TAILED under TEST OF SIGNIFICANCE.

6. Check FLAG SIGNIFICANT CORRELATIONS.

7. Click OK.

SPSS Detailed Steps: Correlation


1. Select ANALYZE from the SPSS menu bar.

2. Click REGRESSION and then LINEAR.

3. Move "Attitude Towards Sports Cars" into the DEPENDENT box.

4. Move "Duration of Car Ownership" into the INDEPENDENT(S) box.

5. Select ENTER in the METHOD box (default option).

SPSS Detailed Steps:Bivariate Regression


6. Click STATISTICS and check ESTIMATES under REGRESSION

COEFFICIENTS.

7. Check MODEL FIT.

8. Click CONTINUE.

9. Click PLOTS.

10. In the LINEAR REGRESSION:PLOTS box, move *ZRESID into the Y:

box and *ZPRED into the X: box.

SPSS Detailed Steps: Bivariate Regression (Cont.)


11. Check HISTOGRAM and NORMAL PROBABILITY PLOT in the

STANDARDIZED RESIDUALS PLOTS.

12. Click CONTINUE.

13. Click OK.

The steps for running multiple regression are similar, except for Step 4.

In Step 4, move "Duration of Car Ownership and "Importance Attached

to Performance" into the INDEPENDENT(S) box.

SPSS Detailed Steps: Bivariate Regression (Cont.)


Excel

Correlations can be determined in EXCEL by using the DATA > DATA

ANALYSIS > CORRELATION function. Use the Correlation

Worksheet Function when a correlation coefficient for two cell ranges

is needed.

Regression can be accessed from the DATA > DATA ANALYSIS

menu. Depending on the features selected, the output can consist of

a summary output table, including an ANOVA table, a standard error

of y estimate, coefficients, standard error of coefficients, R2 and

adjusted R2 values, and the number of observations.


Excel (Cont.)

In addition, the function computes a residual output table,

a residual plot, a line fit plot, normal probability plot, and a

two-column probability data output table.

The detailed steps are illustrated using the data of Table

18.1.


1. Select DATA tab.

2. In the ANALYSIS tab, select DATA ANALYSIS.

3. The DATA ANALYSIS Window pops up.

4. Select CORRELATION from the DATA ANALYSIS Window.

5. Click OK.

6. The CORRELATION pop-up window appears on screen.

7. The CORRELATION window has two portions

1. INPUT

2. OUTPUT OPTIONS

Excel Detailed Steps: Correlation


8. The INPUT portion asks for the following inputs:

a. Click in the INPUT RANGE box. Select (highlight) all the rows of data

under ATTITUDE and DURATION.

$B$2:$C$13 should appear in INPUT RANGE.

b. Select COLUMNS beside GROUPED BY.

c. Leave LABELS IN FIRST ROW as blank.

9. In the OUTPUT OPTIONS window, select NEW WORKBOOK Options.

10. Click OK.

Excel Detailed Steps: Correlation (Cont.)


1. Select DATA tab.

2. In the ANALYSIS tab, select DATA ANALYSIS.

3. DATA ANALYSIS Window pops up.

4. Select REGRESSION from the DATA ANALYSIS Window.

5. Click OK.

6. The REGRESSION pop-up window appears on screen.

Excel Detailed Steps: Bivariate Regression


7. The REGRESSION window has four portions:

a. INPUT

b. OUTPUT OPTIONS

c. RESIDUALS

d. NORMAL PROBABILITY

8. The INPUT portion asks for the following inputs:

a. Click in the INPUT Y RANGE box. Select (highlight) all the

rows of data under ATTITUDE. $B$2:$B$13 should

appear on INPUT Y RANGE.

Excel Detailed Steps: Bivariate Regression (Cont.)


b. Click in the INPUT X RANGE box. Select (highlight) all the rows of

data under DURATION. $C$2:$C$13 should appear on INPUT X

RANGE.

c. Leave LABELS and CONSTANT IS ZERO as blanks.

CONFIDENCE LEVEL should be 95% (default).

9. In the OUTPUT OPTIONS pop-up window, select NEW WORKBOOK

Options.

10. Under RESIDUALS check RESIDUAL PLOTS.

11. Under NORMAL PROBABILITY check NORMAL PROBABILITY PLOTS.

12. Click OK.

Excel Detailed Steps: Bivariate Regression (Cont.)


Excel Detailed Steps: Multiple Regression

The steps for running multiple regression are similar, except for Step 8b. In Step 8b, click in the INPUT X RANGE box.

Select (highlight) all the rows of data under Duration and Importance. $C$2:$D$13 should appear on INPUT X RANGE.


Exhibit 18.1 Other Computer Programs for Correlations

MINITAB

Correlation can be computed using STAT > BASIC STATISTICS > CORRELATION function. It calculates Pearson’s product moment using all the columns.

SASFor a point-and-click approach for performing metric correlations, use the ANALYZE task within SASEnterprise Guide. The MULTIVARIATE > CORRELATIONS task offers Pearson product moment correlations, as well as some other measures.


Exhibit 18.2 Other Computer Programs for Regression

MINITAB

Regression analysis under the STAT > REGRESSION function can perform simple and multiple analysis. The output includes a linear regression equation, table of coefficients R2, R2 adjusted, analysis of variance table, a table of fits and residuals that provide unusual observations and residual plots.

SASFor a point-and-click approach for performing regression analysis, use the ANALYZE task within SASEnterprise Guide. The REGRESSION > LINEAR task calculates bivariate and multiple regression equations, associated statistics, and plots. It allows for an easy examination of residuals.



Acronym: Regression

The main features of regression analysis may be summarized by the acronym REGRESSION:

R esidual analysis is useful

E stimation of parameters: solution of simultaneous equations

G eneral model is linear

R2 strength of association

E rror terms are independent and N(0, s2)

S tandardized regression coefficients

S tandard error of estimate: prediction accuracy

I ndividual coefficients and overall F-tests

O ptimal: minimizes total error

N onstandardized regression coefficientsChapter 11 - 76

Date post:	11-Jan-2016
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18-1 Chapter 18 Data Analysis: Correlation and Regression.

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